A theory for spiral wave drift in reaction-diffusion-mechanics systems

Abstract

Reaction-diffusion mechanics (RDM) systems describe a wide range of practically important phenomena where deformation substantially affects wave and vortex dynamics. Here, we develop the first theory to describe the dynamics of rotating spiral waves in RDM systems, combining response function theory with a mechanical Green's function. This theory explains the mechanically-induced drift of spiral waves as a resonance phenomenon, and it can predict the drift trajectories and the final attractors from measurable characteristics of the system. Theoretical predictions are confirmed by numerical simulations. The results can be applied to cardiac tissue, where the drift of spiral waves is an important factor in determining different types of cardiac arrhythmias.In this work we present an analytical approach to study spiral wave dynamics in RDM systems. We combine response function theory [23] with a Green's function formalism to account for mechanical influence, and we derive an equation for spiral wave drift. The MEF-induced drift is reduced to a resonant forcing problem in the rotating frame of the spiral: during its rotation, the spiral wave perceives a time-varying perturbation since the domain boundaries are not stationary in the spiral's frame of reference. The resonant component of this boundary-induced forcing yields the net spiral drift.Our theoretical predictions are compared to numerical simulations. We find the relative angle and magnitude of the spiral wave's drift and identify the spatial attractors of the system. Some of them, such as the center of the medium, have already been reported in numerical simulations [19]. Using this theory, we also find new regimes and attractors that also are confirmed by simulations. For example, our theory predicts that the center of the domain can also be repulsive, and that multiple stable dynamical attractors may coexist. Although different from previous findings, these predictions are confirmed by numerical simulations.The developed analytical approach allows us to generalize the numerical results, as our analytical findings are based on an Archimedean description of spiral wave geometry, which is common for all types of excitable media. ModelThe reaction and MEF parts of our model for cardiac tissue are as in [18,20,24,25], supplemented here with the Navier-Cauchy equilibrium equations from linear elasticity to facilitate analytical calculations. In a twodimensional medium with Cartesian material coordinates (x, y), the transmembrane voltage u and recovery variable v are evolved according to modified Aliev-Panfilov kinetics [26], as used in [18]:

A survey of pathways for mechano-electric coupling in the atria

“…( Fenton and Karma, 1998 )), coupled with a model for active tension generation ( Brocklehurst et al., 2017 ; Colli Franzone et al., 2017 ; Hu et al., 2013 ; Keldermann et al., 2009 ; Nash and Panfilov, 2004 ; Panfilov et al., 2007 ; Satriano et al., 2018 ; Weise and Panfilov, 2019 ). Findings from these studies corroborate the role of MEC in influencing spiral wave meandering ( Brocklehurst et al., 2017 ; Colli Franzone et al., 2017 ; Dierckx et al., 2015 ; Radszuweit et al., 2015 ) and, in some conditions, spiral wave breakup ( Keldermann et al., 2010 ; Panfilov et al., 2007 ; Weise and Panfilov, 2017 ) and even spiral wave initiation ( Weise and Panfilov, 2011 ; Yapari et al., 2014 ).…”

Cardiac Mechano-Electrical Dynamical Instability

“…However, for G s = 25 S/F MEF causes meandering of the spiral wave on a cycloidal trajectory. This onset of meander can be explained by the resonant drift theory [28] which predicts meandering under a periodical variation of the excitability of the medium, which occurs here due to MEF [23,29]. However, for larger values of G s we observe a hyper-meandering trajectory (see the orange line).…”

Measurement and structure of spiral wave response functions

“…Overlap integrals of spiral wave RFs thus appear in the equations of motion for three-dimensional scroll wave filaments, [13][14][15][16] or in the theory of 2D spiral waves drifting due to a constant external field, 17 surface curvature, 18 or mechano-electrical feedback. 19 In one spatial dimension, the translational RF of the wave front determines the velocitycurvature relation 20,21 and the shape of the RF itself can be used to shape reaction-diffusion patterns. 22 If the underlying model equations are known (e.g., for a cardiac tissue model), the response functions for a (meandering) spiral can be found by linearising the system around its relative equilibrium (periodic orbit).…”